Solutions of Equations in One Variable Newtons Method Numerical - PowerPoint PPT Presentation

Solutions of Equations in One Variable Newton’s Method Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University � 2011 Brooks/Cole, Cengage Learning c

Derivation Example Convergence Final Remarks Outline Newton’s Method: Derivation 1 Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 2 / 33

Derivation Example Convergence Final Remarks Outline Newton’s Method: Derivation 1 Example using Newton’s Method & Fixed-Point Iteration 2 Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 2 / 33

Derivation Example Convergence Final Remarks Outline Newton’s Method: Derivation 1 Example using Newton’s Method & Fixed-Point Iteration 2 Convergence using Newton’s Method 3 Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 2 / 33

Derivation Example Convergence Final Remarks Outline Newton’s Method: Derivation 1 Example using Newton’s Method & Fixed-Point Iteration 2 Convergence using Newton’s Method 3 Final Remarks on Practical Application 4 Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 2 / 33

Derivation Example Convergence Final Remarks Outline Newton’s Method: Derivation 1 Example using Newton’s Method & Fixed-Point Iteration 2 Convergence using Newton’s Method 3 Final Remarks on Practical Application 4 Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 3 / 33

Derivation Example Convergence Final Remarks Newton’s Method Context Newton’s (or the Newton-Raphson ) method is one of the most powerful and well-known numerical methods for solving a root-finding problem. Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 4 / 33

Derivation Example Convergence Final Remarks Newton’s Method Context Newton’s (or the Newton-Raphson ) method is one of the most powerful and well-known numerical methods for solving a root-finding problem. Various ways of introducing Newton’s method Graphically, as is often done in calculus. Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 4 / 33

Derivation Example Convergence Final Remarks Newton’s Method Context Newton’s (or the Newton-Raphson ) method is one of the most powerful and well-known numerical methods for solving a root-finding problem. Various ways of introducing Newton’s method Graphically, as is often done in calculus. As a technique to obtain faster convergence than offered by other types of functional iteration. Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 4 / 33

Derivation Example Convergence Final Remarks Newton’s Method Context Newton’s (or the Newton-Raphson ) method is one of the most powerful and well-known numerical methods for solving a root-finding problem. Various ways of introducing Newton’s method Graphically, as is often done in calculus. As a technique to obtain faster convergence than offered by other types of functional iteration. Using Taylor polynomials. We will see there that this particular derivation produces not only the method, but also a bound for the error of the approximation. Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 4 / 33

Derivation Example Convergence Final Remarks Newton’s Method Derivation Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 5 / 33

Derivation Example Convergence Final Remarks Newton’s Method Derivation Suppose that f ∈ C 2 [ a , b ] . Let p 0 ∈ [ a , b ] be an approximation to p such that f ′ ( p 0 ) � = 0 and | p − p 0 | is “small.” Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 5 / 33

Derivation Example Convergence Final Remarks Newton’s Method Derivation Suppose that f ∈ C 2 [ a , b ] . Let p 0 ∈ [ a , b ] be an approximation to p such that f ′ ( p 0 ) � = 0 and | p − p 0 | is “small.” Consider the first Taylor polynomial for f ( x ) expanded about p 0 and evaluated at x = p . f ( p ) = f ( p 0 ) + ( p − p 0 ) f ′ ( p 0 ) + ( p − p 0 ) 2 f ′′ ( ξ ( p )) , 2 where ξ ( p ) lies between p and p 0 . Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 5 / 33

Derivation Example Convergence Final Remarks Newton’s Method Derivation Suppose that f ∈ C 2 [ a , b ] . Let p 0 ∈ [ a , b ] be an approximation to p such that f ′ ( p 0 ) � = 0 and | p − p 0 | is “small.” Consider the first Taylor polynomial for f ( x ) expanded about p 0 and evaluated at x = p . f ( p ) = f ( p 0 ) + ( p − p 0 ) f ′ ( p 0 ) + ( p − p 0 ) 2 f ′′ ( ξ ( p )) , 2 where ξ ( p ) lies between p and p 0 . Since f ( p ) = 0, this equation gives 0 = f ( p 0 ) + ( p − p 0 ) f ′ ( p 0 ) + ( p − p 0 ) 2 f ′′ ( ξ ( p )) . 2 Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 5 / 33

Derivation Example Convergence Final Remarks Newton’s Method 0 = f ( p 0 ) + ( p − p 0 ) f ′ ( p 0 ) + ( p − p 0 ) 2 f ′′ ( ξ ( p )) . 2 Derivation (Cont’d) Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 6 / 33

Derivation Example Convergence Final Remarks Newton’s Method 0 = f ( p 0 ) + ( p − p 0 ) f ′ ( p 0 ) + ( p − p 0 ) 2 f ′′ ( ξ ( p )) . 2 Derivation (Cont’d) Newton’s method is derived by assuming that since | p − p 0 | is small, the term involving ( p − p 0 ) 2 is much smaller, so 0 ≈ f ( p 0 ) + ( p − p 0 ) f ′ ( p 0 ) . Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 6 / 33

Derivation Example Convergence Final Remarks Newton’s Method 0 = f ( p 0 ) + ( p − p 0 ) f ′ ( p 0 ) + ( p − p 0 ) 2 f ′′ ( ξ ( p )) . 2 Derivation (Cont’d) Newton’s method is derived by assuming that since | p − p 0 | is small, the term involving ( p − p 0 ) 2 is much smaller, so 0 ≈ f ( p 0 ) + ( p − p 0 ) f ′ ( p 0 ) . Solving for p gives p ≈ p 0 − f ( p 0 ) f ′ ( p 0 ) ≡ p 1 . Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 6 / 33

Derivation Example Convergence Final Remarks Newton’s Method p ≈ p 0 − f ( p 0 ) f ′ ( p 0 ) ≡ p 1 . Newton’s Method Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 7 / 33

Derivation Example Convergence Final Remarks Newton’s Method p ≈ p 0 − f ( p 0 ) f ′ ( p 0 ) ≡ p 1 . Newton’s Method This sets the stage for Newton’s method, which starts with an initial approximation p 0 and generates the sequence { p n } ∞ n = 0 , by p n = p n − 1 − f ( p n − 1 ) for n ≥ 1 f ′ ( p n − 1 ) Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 7 / 33

Derivation Example Convergence Final Remarks Newton’s Method: Using Successive Tangents y Slope f 9 ( p 1 ) y 5 f ( x ) ( p 1 , f ( p 1 )) Slope f 9 ( p 0 ) p 0 p p 2 p 1 x x ( p 0 , f ( p 0 )) Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 8 / 33

Derivation Example Convergence Final Remarks Newton’s Algorithm To find a solution to f ( x ) = 0 given an initial approximation p 0 : Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 9 / 33

Derivation Example Convergence Final Remarks Newton’s Algorithm To find a solution to f ( x ) = 0 given an initial approximation p 0 : 1. Set i = 0; 2. While i ≤ N , do Step 3: Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 9 / 33

Derivation Example Convergence Final Remarks Newton’s Algorithm To find a solution to f ( x ) = 0 given an initial approximation p 0 : 1. Set i = 0; 2. While i ≤ N , do Step 3: 3.1 If f ′ ( p 0 ) = 0 then Step 5. Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 9 / 33

Derivation Example Convergence Final Remarks Newton’s Algorithm To find a solution to f ( x ) = 0 given an initial approximation p 0 : 1. Set i = 0; 2. While i ≤ N , do Step 3: 3.1 If f ′ ( p 0 ) = 0 then Step 5. 3.2 Set p = p 0 − f ( p 0 ) / f ′ ( p 0 ) ; Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 9 / 33

Derivation Example Convergence Final Remarks Newton’s Algorithm To find a solution to f ( x ) = 0 given an initial approximation p 0 : 1. Set i = 0; 2. While i ≤ N , do Step 3: 3.1 If f ′ ( p 0 ) = 0 then Step 5. 3.2 Set p = p 0 − f ( p 0 ) / f ′ ( p 0 ) ; 3.3 If | p − p 0 | < TOL then Step 6; Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 9 / 33

Derivation Example Convergence Final Remarks Newton’s Algorithm To find a solution to f ( x ) = 0 given an initial approximation p 0 : 1. Set i = 0; 2. While i ≤ N , do Step 3: 3.1 If f ′ ( p 0 ) = 0 then Step 5. 3.2 Set p = p 0 − f ( p 0 ) / f ′ ( p 0 ) ; 3.3 If | p − p 0 | < TOL then Step 6; 3.4 Set i = i + 1; Numerical Analysis (Chapter 2) Newton’s Method R L Burden & J D Faires 9 / 33